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The Poisson distribution has a simple recurrence relation: given a success rate $\lambda$, the probability of getting $k + 1$ successes in some time $t$ satisfies

$P_{k+1}(t) = \frac{(\lambda t)^{k+1} e^{-\lambda t}}{(k+1)!} = \frac{\lambda t}{k+1} P_{k}(t)$.

I see why the recurrence works algebraically, but is there a nice way to see it intuitively?

kjetil b halvorsen
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Alan O'Donnell
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    There may be many different answers to this. Last year I posted one such answer at https://stats.stackexchange.com/a/215253/919. – whuber Dec 31 '17 at 23:11
  • whuber gives a very detailed answer in the linked question. What one call intuitive can be subjective but there is enough there so that it may satisfy your intuition. – Michael R. Chernick Jan 01 '18 at 00:09
  • The Poisson counts events. I think it is more general to call them events rather than successes. – Michael R. Chernick Jan 01 '18 at 00:22
  • @whuber beautiful! That's exactly what I was looking for. – Alan O'Donnell Jan 01 '18 at 00:35
  • Althought the answer from the other question applies here, I'm not sure if this is really a duplicate - perhaps there are other nice ways to see this recurrence, not directly related to exponential summands – Juho Kokkala Nov 09 '18 at 20:40

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